Problem 3 - Algebra 2 and Trigonometry

Question

In \(3-8,\) for each value of \(\theta,\) use double-angle formulas to find a. \(\sin 2 \theta,\) b. \(\cos 2 \theta,\) c. \(\tan 2 \theta .\) Show all work. $$ \theta=30^{\circ} $$

Step-by-Step Solution

Answer

\(\sin 2\theta = \frac{\sqrt{3}}{2}, \cos 2\theta = \frac{1}{2}, \tan 2\theta = \sqrt{3}\).

Step 1: Identify Double-Angle Identities

For double-angle identities, we use the following formulas: \(\sin 2\theta = 2\sin\theta\cos\theta\), \(\cos 2\theta = \cos^2\theta - \sin^2\theta\), and \(\tan 2\theta = \frac{2\tan\theta}{1 - \tan^2\theta}\).

Step 2: Calculate \( \sin\theta \) and \( \cos\theta \) for \( \theta = 30^{\circ} \)

For \( \theta = 30^{\circ} \), we use the known trigonometric values: \( \sin 30^{\circ} = \frac{1}{2} \) and \( \cos 30^{\circ} = \frac{\sqrt{3}}{2} \).

Step 3: Apply Double-Angle Formula for \( \sin 2\theta \)

Using \( \sin 2\theta = 2\sin\theta\cos\theta \), substitute \( \sin 30^{\circ} = \frac{1}{2} \) and \( \cos 30^{\circ} = \frac{\sqrt{3}}{2} \): \[ \sin 2(30^{\circ}) = 2 \left(\frac{1}{2}\right) \left(\frac{\sqrt{3}}{2}\right) = \frac{\sqrt{3}}{2} \].

Step 4: Apply Double-Angle Formula for \( \cos 2\theta \)

Using \( \cos 2\theta = \cos^2\theta - \sin^2\theta \), substitute \( \cos 30^{\circ} = \frac{\sqrt{3}}{2} \) and \( \sin 30^{\circ} = \frac{1}{2} \): \[ \cos 2(30^{\circ}) = \left(\frac{\sqrt{3}}{2}\right)^2 - \left(\frac{1}{2}\right)^2 = \frac{3}{4} - \frac{1}{4} = \frac{1}{2} \].

Step 5: Find \( \tan\theta \) and Apply Double-Angle Formula for \( \tan 2\theta \)

First, calculate \( \tan 30^{\circ} = \frac{\sin 30^{\circ}}{\cos 30^{\circ}} = \frac{1/2}{\sqrt{3}/2} = \frac{1}{\sqrt{3}} \). Then use the formula \( \tan 2\theta = \frac{2\tan\theta}{1-\tan^2\theta} \):\[ \tan 2(30^{\circ}) = \frac{2\left(\frac{1}{\sqrt{3}}\right)}{1-\left(\frac{1}{\sqrt{3}}\right)^2} = \frac{\frac{2}{\sqrt{3}}}{1-\frac{1}{3}} = \frac{\frac{2}{\sqrt{3}}}{\frac{2}{3}} = \sqrt{3} \].

Key Concepts

Understanding Trigonometric Identities

Trigonometric identities are essential formulas that help simplify complex trigonometric expressions and equations. They emerge from the inherent relationships between the angles and lengths of triangles. These identities are useful in calculus, physics, engineering, and many areas of science.
Some of the most frequently used trigonometric identities are the Pythagorean identities, angle sum and difference identities, and the double-angle identities. The double-angle identities are:

• \( \sin 2\theta = 2\sin\theta\cos\theta \)
• \( \cos 2\theta = \cos^2\theta - \sin^2\theta \)
• \( \tan 2\theta = \frac{2\tan\theta}{1 - \tan^2\theta} \)

Using these identities can simplify calculations when dealing with angles double those we initially know or use. Knowing these can significantly aid in solving trigonometric problems involving double angles.

Evaluating Sine and Cosine Values

Sine and cosine are fundamental trigonometric functions that are associated with the ratios of sides in a right triangle. When dealing with a specific angle like \(30^{\circ}\), we can readily identify their values, which are used frequently. This is because certain angles have known sine and cosine values due to their occurrence in special triangles such as the 30-60-90 triangle.
The sine and cosine of \(30^{\circ}\) are:

• \(\sin 30^{\circ} = \frac{1}{2}\)
• \(\cos 30^{\circ} = \frac{\sqrt{3}}{2}\)

These values are derived from the geometric properties of the equilateral triangle, where dropping a perpendicular from one vertex splits it into two 30-60-90 triangles. Understanding these values helps in simplifying calculations involving trigonometric functions for various angles.

Applying Tangent Identities

The tangent function is another pivotal trigonometric function defined as the ratio of sine over cosine. For any angle \(\theta\), the formula is \(\tan\theta = \frac{\sin\theta}{\cos\theta}\).
When using the double-angle identity for tangent, \(\tan 2\theta\), we use:

• \( \tan 2\theta = \frac{2\tan\theta}{1 - \tan^2\theta} \)

Let's consider \(30^{\circ}\):

• First, calculate \(\tan 30^{\circ} = \frac{\sin 30^{\circ}}{\cos 30^{\circ}} = \frac{1/2}{\sqrt{3}/2} = \frac{1}{\sqrt{3}}\).
• Then apply the double-angle formula: \(\tan 2(30^{\circ}) = \frac{2\left(\frac{1}{\sqrt{3}}\right)}{1-\left(\frac{1}{\sqrt{3}}\right)^2} = \sqrt{3}\).

By understanding how to combine these basic trigonometric values with the identities, especially for double angles, you can solve complex trigonometric problems with greater ease.